LINEAR MODELS AND MATRIX ALGEBRA Chapter 4 Alpha Chiang, Fundamental Methods of Mathematical Economics 3rd edition

Why Matrix Algebra As more and more commodities are included in models, solution formulas become cumbersome. Matrix algebra enables to do us many things: provides a compact way of writing an equation system leads to a way of testing the existence of a solution by evaluation of a determinant gives a method of finding solution (if it exists)

Catch Catch: matrix algebra is only applicable to linear equation systems. However, some transformation can be done to obtain a linear relation. y = axb log y = log a + b log x

Matrices and Vectors Example of a system of linear equations: c1P1 + c2P2 = -c0 1P1 + 2P2 = -0 In general, a11 x1 + a12 x2 +…+ a1nXn = d1 a21 x1 + a22 x2 +…+ a2nXn = d2 ……………………………… am1 x1 + am2 x2 +…+ amnXn = dm coefficients aij variables x1, …,xn constants d1, …,dm

Matrices as Arrays

Example: 6x1 + 3x2+ x3 = 22 x1 + 4x2+-2x3 =12 4x1 - x2 + 5x3 = 10

Definition of Matrix A matrix is defined as a rectangular array of numbers, parameters, or variables. Members of the array are termed elements of the matrix. Coefficient matrix: A=[aij]

Matrix Dimensions Dimension of a matrix = number of rows x number of columns, m x n m rows n columns Note: row number always precedes the column number. this is in line with way the two subscripts are in aij are ordered. Special case: m = n, a square matrix

Vectors as Special Matrices one column : column vector one row: row vector usually distinguished from a column vector by the use of a primed symbol: Note that a vector is merely an ordered n-tuple and as such it may be interpreted as a point in an n-dimensional space.

Matrix Notation Ax = d Questions: How do we multiply A and x? What is the meaning of equality?

Example Qd = Qs Qd = a - bP Q s= -c + dP can be rewritten as 1Qd + bP = a 0 +1Qs +-dP = -c

In matrix form… Constant vector Coefficient matrix Variable vector

Matrix Operations Addition and Subtraction: matrices must have the same dimensions Example 1: Example 2:

Matrix addition and subtraction In general Note that the sum matrix must have the same dimension as the component matrices.

Matrix subtraction Subtraction Example

Scalar Multiplication To multiply a matrix by a number – by a scalar – is to multiply every element of that matrix by the given scalar. Note that the rationale for the name scalar is that it scales up or down the matrix by a certain multiple. It can also be a negative number.

Matrix Multiplication Given 2 matrices A and B, we want to find the product AB. The conformability condition for multiplication is that the column dimension of A (the lead matrix) must be equal to the row dimension of B ( the lag matrix). BA is not defined since the conformability condition for multiplication is not satisfied.

Matrix Multiplication In general, if A is of dimension m x n and B is of dimension p x q, the matrix product AB will be defined only if n = p. If defined the product matrix AB will have the dimension m x q, the same number of rows as the lead matrix A and the same number of columns as the lag matrix B.

Matrix Multiplication Exact Procedure

Matrix multiplication Example : 2x2, 2x2, 2x2

Matrix multiplication Example: 3x2, 2x1, 3x1

Matrix multiplication Example: 3x3, 3x3, 3x3 Note, the last matrix is a square matrix with 1s in its principal diagonal and 0s everywhere else, is known as identity matrix

Matrix multiplication from 4.4, p56 The product on the right is a column vector

Matrix multiplication When we write Ax= d, we have

Simple national income model Example : Simple national income model with two endogenous variables, Y and C Y = C + Io + Go C = a + bY can be rearranged into the standard format Y – C = Io – Go -bY + C = a

Simple national income model Coefficient matrix, vector of variables, vector of constants To express it in terms of Ax=d,

Simple national income model Thus, the matrix notation Ax=d would give us The equation Ax=d precisely represents the original equation system.

Digression on  notation: Subcripted symbols helps in designating the locations of parameters and variables but also lends itself to a flexible shorthand for denoting sums of terms, such as those which arise during the process of matrix multiplication. j: summation index xj: summand

Digression on  notation:

Digression on  notation: The application of  notation can be readily extended to cases in which the x term is prefixed with a coefficient or in which each term in the sum is raised to some integer power. general polynomial function

Digression on  notation: Applying to each element of the product matrix C=AB

Digression on  notation: Extending to an m x n matrix, A=[aik] and an n x p matrix B=[bkj], we may now write the elements of the m x p matrix AB=C=[cij] as or more generally,